In the book [Lectures on curves on an algebraic surface] by Mumford, Example 3 on p.2 says the projections $C\to \mathbb{P}^1$, $C\to E$ are covering maps where $E$ is an elliptic curve over $k$ and $C$ is a curve on $\mathbb{P}^1\times E$.
Does this mean the underlying topological maps of $C\to \mathbb{P}^1$, $C\to E$ are covering maps in the sense of topology? If it does, may I ask why?